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Kinetic roughening of a randomly growing surface can be modelled by theKardar-Parisi-Zhang equation with a time-independent (``spatially quenched'' or``columnar'') random noise. In this paper, we use the field-theoreticrenormalization group approach to investigate how randomly moving mediumaffects the kinetic roughening. The medium is described by the stochasticdifferential Navier-Stokes equation for incompressible viscous fluid with anexternal stirring force. We find that the action functional for the fullstochastic problem should be extended to be renormalizable: a new nonlinearitymust be introduced. Moreover, in order to correctly couple the scalar andvelocity fields, a new dimensionless parameter must be introduced as a factorin the covariant derivative of the scalar field. The resulting actionfunctional involves four coupling constants and a dimensionless ratio ofkinematic coefficients. The one-loop calculation (the leading order of theexpansion in $\varepsilon=4-d$ with $d$ being the space dimension) shows thatthe renormalization group equations in the five-dimensional space of thoseparameters reveal a curve of fixed points that involves an infrared attractivesegment for $\varepsilon>0$.

Effect of random environment on kinetic roughening: Kardar-Parisi-Zhang model with a static noise coupled to the Navier-Stokes equation